Dynamical Systems and Group Actions.

Intro -- Preface -- Bibliography -- Hyperfinite actions on countable sets and probability measure spaces -- 1. Introduction -- Acknowledgement -- 2. The Stone-Cech compactification -- 3. Geometrically hyperfinite actions -- 4. Graphs and graphings -- 5. On faithfulness -- 6. On a problem of van Douwen -- 7. A topologically free, hyperfinite action of a nonamenable group -- References -- A road to the spectral radius of transfer operators -- 1. Introduction -- 2. Reversible dynamics. Spectral radius and ergodic measures -- 3. Irreversible-reversible dynamics -- 4. Irreversible dynamics. Weighted shifts and Perron-Frobenius operators. Spectral radius and entropy -- 5. Spectral potential of a transfer operator. Convexity, Legendre duality and thermodynamics -- 6. Spectral potentials and dual entropy -- 7. -entropy and variational principle for transfer operators -- 8. Properties of -entropy. Entropy Statistic Theorem and Variational Principle for -entropy -- 9. Variational principle for weighted shift operators -- 10. Multiterm functional operators. Variational principles etc. -- References -- A condition for weak mixing of induced IETs -- 1. IETs: Minimality, ergodicity and mixing -- 2. The results -- 3. Some notation, terminology and lemmas -- 4. Proof of Theorem 3 -- 5. Proof of Theorem 2 -- 6. Proof of Proposition 3 -- 7. Two extensions of Theorem 1 -- 8. Final comments -- References -- Every countably infinite group is almost Ornstein -- 1. Introduction -- 2. Preliminaries -- 3. Almost Ornstein groups -- 4. Measurable subgroups -- 5. Ornstein groups -- References -- Fair dice-like hyperbolic systems -- 1. Introduction -- 2. Definitions and main results -- Concluding remarks -- References -- Complexity and heights of tori -- 1. Introduction -- 2. Spectral preliminaries for discrete tori -- 3. Spectral preliminaries for continuous tori.

4. Asymptotics -- 5. Proof of the corollaries -- References -- Flows with uncountable but meager group of self-similarities -- 0. Introduction -- 1. Gaussian examples -- 2. Poisson flows -- 3. Concluding remarks and problems -- References -- The universal minimal space of the homeomorphism group of a h-homogeneous space -- 1. Introduction -- 2. Preliminaries -- 3. Basic properties of h-homogeneous spaces -- 4. Calculation of the universal minimal space -- References -- Generic eigenvalues, generic factors and weak disjointness -- 1. Introduction -- 2. Generic eigenvalues and weakly scattering -- 3. Generic homomorphisms, generic factors and weak disjointness -- 4. Relations among several scattering properties -- 5. Scattering properties in an almost equicontinuous system -- References -- Around King's rank-one theorems: Flows and ℤⁿ-actions -- 1. Introduction -- 2. Preliminaries and notations -- 3. Rank-one flows -- 4. Approximation theorem -- 5. Weak closure theorem for rank-one flows -- 6. Rigidity of factors of rank-one flows -- 7. King's theorem for flat-roof rank-one flow -- 8. ℤⁿ-Rank-one action -- References -- Random walks on random horospheric products -- Introduction -- 1. Asymptotic geometry of horospheric products -- 2. Random horospheric products -- References -- Statistics of gaps in the sequence {√ } -- Dedication -- 1. Formulation of the Problem -- 2. Preliminary construction -- 3. The analysis of the differences √ _{ ₁, }-√ _{ ₂, } -- 4. The case of arbitrary ₁, ₂. -- 5. The remainders -- References -- Invariant distributions for interval exchange transformations -- 1. Introduction -- 2. The spaces ℒ( ) and ( ) -- 3. Finitely additive invariant measures and coboundaries -- 4. ℒ( ) and invariant distributions -- 5. Zippered rectangles flow -- 6. Multiplicative coboundaries -- 7. The Kontsevich-Zorich cocycle.

8. The Rauzy cocycle -- References -- Notes on the Schreier graphs of the Grigorchuk group -- 1. Introduction -- 2. Space of marked graphs -- 3. Group actions -- 4. The Schreier graphs -- 5. Space of subgroups -- 6. Automorphisms of regular rooted trees -- 7. The Grigorchuk group -- References -- Minimal models for free actions -- 1. Introduction -- 2. The construction of the _{ } -- 3. Invariant measures on the 1-acceptable sequences -- 4. Constructing the partition -- 5. Almost 1-1 extensions -- 6. Minimal models for free actions -- References.

This volume contains cutting-edge research from leading experts in ergodic theory, dynamical systems and group actions. A large part of the volume addresses various aspects of ergodic theory of general group actions including local entropy theory, universal minimal spaces, minimal models and rank one transformations. Other papers deal with interval exchange transformations, hyperbolic dynamics, transfer operators, amenable actions and group actions on graphs.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.